The so-called "proofs" for the existence of God are not proofs in a mathematical sense. They are more along the lines of evidences, or clues, or intuitions, that suggest or point to the existence of God, and which make the existence of God a reasonable and justified (or warranted) belief.
Anselm's Ontological argument is one of the most unusual. It has been hotly contested for nearly 1,000 years. It was attacked by Kant and defended by Hegel. More recently, Alvin Plantinga is a 21st cc. defender of the ontological argument.
Here it is, in summary form (this is my paraphrase -- Anselm scholars can tell me if I have represented him accurately):
1. God exists in human understanding as the greatest of all possible beings. (Even "the fool" who says in his heart "there is no God" has a definite sense of the being whose existence he is denying -- a supreme, greatest being for whom a greater being is, by definition, neither possible nor conceivable.)
2. The concept of God as the greatest possible being resides in our minds and is not an incoherent or nonsensical category of thought (unlike, e.g., flying spaghetti monsters, for whom atheists seem to have a particularly unhealthy obsession). *Update: see second comment, below*.
3. Something that exists only in our understanding is not as great as something that exists in reality. If God exists only in our minds but not in reality, it would in fact be possible to conceive of a greater being -- one who exists in reality.
4. Therefore, it would be contradictory to hold in our minds the concept of the greatest possible being and to deny his existence.
5. Therefore God, the greatest possible being, exists in reality as well as our understanding.
The argument is notable because it is the only one that seeks to prove the existence of God on the basis of pure reason without recourse to perception or empirical evidence. The proof relies on a human being's seemingly innate and pre-wired mental capacity to intuitively grasp the concept of God as the greatest of all possible beings.